I'm in a lunch group at work of recreational math geeks and we came up with a question which we need help to resolve. I apologize in advance, if my explanation is not perfectly rigorous.
Given these two statements:
- Between any two rational number X and Y there is an irrational number
- Between any two irrational numbers X and Y there is a rational number
- Rational numbers are countably infinite and irrationals are uncountably infinite
Does that imply that the real numbers always alternate between the rationals and irrationals (i.e. rational -> irrational -> rational -> irrational...etc) no matter how close X and Y get?
It seems like the vastly larger number of irrationals implies that there are two irrationals that don't have a rational between them.
What can help us resolve this misunderstanding on our part?